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47Notre Dame Journal of Formal LogicVolume X, Number 1, January 1969

THE NUMERICAL EPSILON

JOHN THOMAS CANTY

In this paper* Lesniewski's system of ontology extended by an axiom ofinfinity is used to derive Peano's arithmetic. Section 1 gives the maintheses of this derivation which parallels the work of [6]. Using thenumerical epsilon, defined in section 2, Peano's arithmetic is given acharacteristically ontological model in section 3. Thus, the paper provides,for Peano's arithmetic, the two ways of treating logical concepts inontology, the one, protothetical (section 1), the other, ontological (section 3).

1. Numerals as predicates The following proposition, in which the epsilonis primitive and is a proposition forming functor for two name arguments,is taken as the single axiom of ontology and is understood to be added tosome given development of protothetic.

[Ab] Λ Aεb.=: [3C].CεA:[C]:CεA.z>.Cεb:[CD]:CεA.DεA.Ό.CεD

There is no rule which determines the style of letters to be used forvariables, but throughout the paper capital Latin letters will be used forproper name variables and lower case Latin letters for general namevariables; Greek letters will be employed for variables of higher semanti-cal categories. Two types of definition, with the usual restrictions forbound and free variables, are allowable in ontology; ontological definitionswhich have the form:

[Aabc . . .]:[36].Aε5.Φ(Aα6c . . .) .=.Azτ<abc . . . >

and protothetical definitions which are of the form:

[abc . . ,]:Φ(abc . . ,).=.τ{abc . . .)

*This paper is part of a Thesis written under the direction of Professor BoleslawSobociήski and submitted to the Graduate School of the University of Notre Dame, inpartial fulfillment of the requirements for the degree of Doctor of Philosophy withPhilosophy as major subject in June, 1967.

Received August 1, 1967

48 JOHN THOMAS CANTY

where τ is the constant being defined and is a name forming functor in thefirst case and a proposition forming functor in the second. The latter type ofdefinition is called protothetical since, although the directive for ontologyexplicitly allows it as an acceptable form of definition, it is also allowableas a form of definition in Lesniewski's system of protothetic.

Once a constant of a given semantical category is introduced into thesystem, the principle of extensionality for that category is allowable as athesis of the system. A more complete explanation of the directives forontology and protothetic is found in [l], [4] and [5].

Two consequences of this axiom,

T1.2 \Ab\\Azb .-D.AzA

T1.3 \ABc]:AεB. Bεc .Ό.Aεc

serve to illustrate the difference between the primitive epsilon of ontologyand other epsilons (of, say, set theory), since the former is seen to besemi-reflexive and transitive.

A list of well known basic definitions and theses follows, many ofwhich can be found in [2].

D1Λ [a]:[3A].Aεa.=. I {a}D1.2 [σ].i.[BC]:£εc.Cεα.D.fiεC:=.-»{c}T1.4 [A]:l{A}.->{A}.=.AεAD1.3 [A]:AεA.=.AεVD1.4 [A]:AεA.~(AεA).=.AεΛD1.5 [AB]:AεB.BεA.=.A=BD1.6 [AB]:AεA.BεB.~(A=B).=.AίBD1.7 [ab].'. [A] :Aεa .=.Aεb :=.aobT1.5 [ab]:. [A] :Aεa .=.Aεb:=: [φ]:φ{a}.=. φ{b}T1.6 [ab):.aqb ,=:[φ]: φ{a}.=. φ{b}D1.8 [Aφ]:AεA.φ{A}.=.Aεstsi<φ>T1.7 [AB]:.A=B.=:AεA.BεB:[φ]:φ{A}.=.φ{B}D1.9 [ab]:.[A]:Aεa.'D.Aεb:=.atbD1.10 [Aab]:Aεa.Aεb.=.Aεaf)bDl.ll [Aab].'.AεA: Aεa .v.Aεb :=.AεaUbT1.8 [Aab]:.Aεa.v.Aεb:=.AεaUbT1.9 [Aa]:Aεa.^.a\jAoaD1.12 [Aab]:Aεa.~(Aεb).=.Aεa-bT1.10 [Aa]:AεA.~(iAεa).^.(auA)-AoaD1.13 [flδ]:βΠδoΛ.Ξ.αV6

D1.14 [φ]:[3a]φ{a}.=. Uφ)D1.15 [φψμ]:ψ(φ).μ(φ).^n<ψμ>(φ)D1.16 [φψ]:.[a]:φ{a}.'D.ψ{a}:=.φc:ψD1.17 [φψ]:/[a]:φ{a}.=.ψ{a}:=. φoψTl.ll [φψ]:.[a]:φ{a}.=.ψ{a}:^[μ]:μ(φ).=. μ(ψ)T1.12 [φψ]:.φoψ.=:[μ]:μ(φ).=. μ{ψ)

As can be seen from the above, there are many parallels between

THE NUMERICAL EPSILON 49

ontology and other systems, for instance [6], though in onotlogy one obtains

two analogues of each logical concept, one ontological, the other proto-

thetical, see for instance, Dl.l with D1.14 and D1.9 with D1.16. Indeed, one

obtains a definition of cardinal number from that of equinumerosity as in

[6], and in section 3 an ontological, rather than protothetical, development

of this theory is given.

D1.18 [ab].'.[lφ]:[ABC]: φ{ΛCJ. φ{BC} .O.A = B:[ABC]: φ{CA}. φ{CB} .Z).

A =B : [A] :Aεa.=. [IB]. φ{BA} : [B] : Bε b .=. [3A]. φ{BA} :=. a oob

T1.13 [a].a*>a

T1.14 [ab] :a oob .D. b <*>a

T1.15 [abc] :aoob .b ooc ."D.a °oc

D1.19 [ab]:aoob.=.co<a>{b]

T1.16 [a\.oo<a >{a}

D1.20 [φ].' .[ab]: φ{a}.aoob.^. φ{b}:=.H(φ)

T1.17 [α].N(oo<α>)

D1.21 [φ].'.[ab]: φ{a}. φ{b} .D. a oob :=.Q(φ)

T1.18 [α].Q(oo<α>)

D1.22 [φ]:l(φ)M(φ).Q(φ).^HC(φ)

T1.19 [α].NC(oo<α>)

Definitions D1.20 and D1.21 provide the concepts of a proposition

forming functor (for one name argument) being numerical and quantitative.

These concepts, together with the condition that a given functor is unempty,

provide the defining characteristics for a cardinal number—see D1.22.

Finite names, zero, and the successor function are also definable,

though the definition of successor given here is that of Frege rather than

that of [6]. The definition of successor in [6] amounts to a particular case

of addition which is defined as the union of disjoint sets. However, this

definition is ambiguous as to type (the successor of a number is not

necessarily of the same type as the number) and is therefore not as good a

candidate for use in ontology as the Fregean definition. Of course the

Fregean definition is derivable as a theorem of [6]—see *110.63.

D1.23 [a]::[φ].\ φ{λ} : [Ab]: Aεa . φ{b} .3 . φ{b\jA}:Z). φ{a}.\ =. Fin {4

T1.20 Fin{Λ}

T1.21 [Aa]:AεA.Fin{a}.Z).Fm{a{jA}

T1.22 N(Fin)

T1.23 [αδ]:Fin{α}.δcα.~(αcδ).D. ~{a oob)

D1.24 [a]:aoA.=.θ{a}

T1.24 θ{Λ}

T1.25 [aφ]:θ{a}.φ{a}.^.φ{Λ}

T1.26 N(0)

T1.27 Q(0)

D1.25 [aφ]:[lA].Aεa.φ{a-A}.=.S<φ>{a}

T1.28 [φ].~([la].θ{a}.S< φ>{a})

T1.29 [aφψ] :N(φ) .S<φ>{a} .S<ψ>{a} .D. [36]. φ{b} . ψ{b}


T1.30 [φ]:U{φ).^M(S<φ>)

T1.31 [φ]:Q(φ).^.Q(S<φ>)-

The following proposition is taken as the axiom of infinity,

[a]: Fπφ}.D. [3A].AεA.~(Aεa)

and given this axiom it is now possible to derive Peano's axioms. First,one defines natural number,

D1.26 [Φ]:: [ μ ] . \ μ(0): [Ψ]: μ(ψ) .D. μ ( S < ψ » : D . μ(φ).' =.Hn(φ)

and then the following are derivable.

T1.32 Nn(0)T1.33 [<ρ]:Nn(<ρ).D.~(0oS<<p>)

- T1.34 [<p]:Nn(<p).D.Nn(S<<p>)T1.35 [φμ].'. μ(0): [Ψ]: Nn(ψ) . μ(ψ) .D. μ ( S < ψ » : Nn(<p) :D. μ(<p)77.36 [<ρ]:Nn(<ρ) .D.<ρc FinΓ2.57 NncNC77.3S [^ψ]:Nn(φ).Nn(^/).S<(^>oS<ψ>.D.^oι//

Of these theses, T1.32—T1.35 and T1.38 are Peano's axioms and T1.34follows in the system banally since its consequent is a thesis, while onlyT1.38 requires the axiom of infinity for its proof.

Given these axioms, it is possible to obtain the rest of Peano'sarithmetic by defining the various arithmetical operations and derivingtheir consequences. The approach here is to use Frege's method ofreducing implicit definitions to explicit ones using "impredicative" defini-tions. For instance, guided by the recursive definition of addition, onedefines,

Ώ1.27 [T] .*. [φ]: Hn(φ) . 3 . τ{φφθ) :=. sm(τ)D1.28 [τ].*.[^ψμ]:Nn(^).Nn(ψ).Nn(μ).τ(^ψμ).D.

(S<φ>ψS< μ>):=.smm(τ)D1.29 [φψμ].'~Nn{φ).Nn(ι//).Nn(μ): [r] : sm(τ). smm(τ) . D .

τ(φψ.μ):=.Sm{φψμ)

and then obtains,

T1.39 [<p]:Nn(<ρ).=>.Sm(<ρ<ρO)T1Λ0 [ωψμ]: Sm(φψμ) .^.Sm(S<φ>ψS< μ>)T1A1 [^]:Nn(<^).Nn(ψ) .=). [3μ].Nn(μ).Sm(μ^)T1A2 [φψμr]:Sm(φμτ) .Sm(ι//μτ) .Ώ.φoψ

Now, given the definition of addition,

D1.30 [aφΨ]:Sm(°o<a>φψ) =.+<φψ> {a}

the uniqueness, closure and recursive properties of the operation follow.

T1A3 [φψμ].'Mn(φ) .Ό:Sm(φψμ) .=. φoψ+μT1A4 [φψ]:Mn(φ) .Hn{ψ) .^.Un(φ + ψ)


T1A5 [φ]:Nn(φ) .D. <p + 0o φT1.46 [φψ]:Hn(φ) .Hn(ψ) .=). φ + S< ψ>oS< φ+ψ>

Clearly, all recursively definable concepts are obtainable in thismanner. But it is important to note that this is probably the only way theyare obtainable in this system. For instance, one cannot use the concept ofordered pairs in order to define multiplication as is done in [6]. Forthough ordered pairs are definable in ontology, an ordered pair of argu-ments is of a different semantical category than its arguments. Thisimmediately leads to the problem of having non-homogeneous argumentsfor arithmetical operations—a problem which cannot be ignored in ontologybecause the system does not allow "typical ambiguity" (as does the systemof [6]): any variable appearing in a thesis is definite as to type in ontology.

However, it is clear that Peano's arithmetic is derivable in thissystem in a way at least similar to [6]. Numerals are construed aspredicates, that is, proposition forming functors for names, and numeralsfor finite numbers are characteristically inductive.

2. The numerical epsilon Consider the following definition of a higherepsilon.

D2Λ [ΦφΨ].'. [la].Φ{a}.φ{a}: [ab]: Φ{a}.Φ{b}.^.ψ{ab}: [ab]:Φ{a}.ψ{ab}.D.Φ{b}: [ab]: φ{a\. ψ{ab}.D. φ{b}:=. ε <ψ>(Φ<p)

In what follows interest will be limited to the particular case,

T2.1 [Φφ].'.[la].Φ{a}. φ{a}: [ab]:Φ{a}. Φ{b}.^.aoob: [ab]:Φ{a}.a<χ>b .Z).Φ{b}: [ab]: φ{a}.aoo b .D. φ{b}:=. ε<°o >(Φφ)

and hereafter 'Φεoo<p' will be written for 'ε <oo>(Φ<ρ)\ This higherepsilon will be called a numerical epsilon and could have been defined bythe following thesis.

T2.2 [Φφ]: [la]. Φ {a}. φ{a}. Q(Φ). N(Φ). N(φ) .=. Φε^φ[T2.1, D1.20, D1.21]

In [l] the author gave T2.2 as the definition of his numerical epsilon,but it is more perspicuous to view the definition as merely one instance of ageneral form for defining higher epsilons. For instance, D2.1 also yields

T ε < o > [Φφ].'. [la].Φ{a}.φ{a}: [ab]: Φ{a}. Φ{b} .^.aob : [ab]:Φ{a\.aob .D.Φ{δ}: [ab]: φ{a}.aob .D. φ{b}:=. ε <o>(Φ(p)

which is equivalent to the well known definition of a higher epsilon

D ε < o > [Φφ].'. [la]. Φ{a}. φ{a}: [ab]: Φ{a}. Φ{b}.^).aob :=. ε <o>(Φ(p)

since in this case T ε < o > contains subformulas, which by the principle ofextensionality are theses (see T1.6). Similarly one would have

T ε < = > [Φφ].'.[lA].Φ{A}.φ{A^:[AB]:Φ{A}.φ{B}.D.A=B:[AB]:Φ{A}.A=B.Ό.Φ{B}: [AB]: φ{A}.A =B .D. φ{B}:=. ε <=>(Φ^)


and equivalently

Dε<=> [Φφ].'.[lA].Φ{A}.φ{A}:[AB]:ΦltA}.Φ{$.D.A=B:=.ε< = >(Φφ)

which is the usual definition given for the restricted higher epsilon. Thisgeneralization and particularly its use for defining a numerical epsilon asopposed to a higher general or restricted epsilon marks the author'scontribution to the theory of higher epsilons in the field of ontology. Aswill be seen, the numerical epsilon is useful since it allows the construc-tion of a characteristically ontological model for numerical concepts.

Firs t of all, on the basis of D2.1 it is possible to derive a thesisemploying the numerical epsilon which is analogous to the axiom ofontology (which uses the primitive epsilon of ontology).

T2.3 [ΦφY.Φεcoφ.Ό.ΦεooΦ [T2.2]

T2A [ΦΦφ]: Φε^φ. ΨεooΦ .^.^Coo^PF: [Φ*<p]ΛHyp(2).D:

3) Q(Φ)M(φ). [T2.2,l]4)Q(Φ).N(Φ): [T2,2,2]

[3β]:5) Φ{a}.φ{a}. [T2.2, l ]

[36].6) *'{δ}.Φ{&} [T2.2,2]7) aoob [D1.21, 3, 5, β]8) φ{b}: [D1.20, 3, 5, 7]

Vεvoψ [T2.2, 3, 4, 6, 8]

In the demonstration of this thesis the first line abbreviates listing thehypotheses of the conditional to be proved and the numeral indicates thenumber of components in the antecedent of the conditional.

T2.5 [ΦΨXφ]: Φε oo(/?. Φε ooΦ . Xε ooΦ . D . Φε ooX

PF: [ΦΨX<p].\Hyp(3).D:4) Q(Φ) [T2.2, 1]5) Q(t t ) .N(¥). [T2.2,2]6)N(X): [T2.2,2]

[3«]:7) *{a}.Φ{a}. [T2.2,2]

[35].8) X{b}.Φ{b}. [T2.2,Z]9) aoob . [D1.21, 4, 7, 8]

10) ^{6}: [D1.20, 5, 7, 9]Ψε^X [T2.2, 10, 8, 5, 6]

T2.6 [aφ]: φ{a}.N(φ) .=.<*><α> ε^φ[T2.2, D1.19, T1.16, T1.17, T1.18, T1.14]

T2.7 [abφ].\a°°b .D: oo<α> ε^φ.^. °o<6> ε^φ [T1.14, T2.6]T2.8 [ΦΨabj.'.^ε^Φ: [ΨXy.Vε^Φ .Xε^Φ.D. ΨεooX: Φ{a}. Φ {δ}:D.α <*>&


PF: [ΦΦaδ].\Hyp(4):D.

5) N(Φ). [T2.2, 1]

6) °o< α >εooΦ. [T2.6, 3, 5]

7) ° o < δ > ε o o Φ . [T2.6, 4, 5]

8) oo<a> ε o o°o<5> . [2, 6, 7]

[3c].

9) o o < β > { c } . o o < δ > {c}. [T2.2, 8]

aoob [D1.19, 9, T1.14, T1.15]

T2.9 [ΦΦ]. .ΦεooΦ:[ΦX]:ΦεooΦ.XεooΦ.D.ΦεooX:D.Q(Φ) [Γ2.S, Z>2.£2]

Γ2.20 [ΦΦ<ρ].'.ΦεooΦ: [ΦXj .ΦεooΦ.XεooΦ.D.ΦεooX: [Φj .ΦεooΦ.D.

Φεoo(^:=).Φεoo^

PF: [ΦΦ(p].\Hyp(3):z>.

4) N(Φ). [T2.2, 1]

5) Q(Φ). [T2.9, 1,2]

6) Φ{α} [Γ^.2, 1]

7) ΦεooΦ. [T2.2, 6, 5, 4]

Φεooc^ [3,7]

T2.11 [ Φ ^ . ' . Φ ε o o ^ . s : [3Φ].ΦεooΦ: [ΦX]: ΦεooΦ .XεooΦ . 3 .

Φε^X: [Φj .ΦεooΦ.D.Φεco^ [T2.10, T2.3, T2.4, T2.5]

This thesis is the analogue of the axiom of ontology since it can be

expressed as a proposition identical in shape with the axiom except for the

parentheses which are generally associated with the primitive epsilon on

the one hand and the numerical epsilon on the other.

Under certain conditions, the principle of extensionality holds for the

numerical epsilon. The limitation imposed on employing extensionality for

the numerical epsilon amounts to restricting the principles's application to

numerical predicates. This is clear when one considers T2.14. On the

basis of the auxiliary definition,

D2.2 [Φφ]: Φεoo(p.=. εoo<Φ>(<?)

there follows,

T2.12 [φψ].\[μ]> β(Φ) Ξ β(Ψ) :=>: [Φ]: Φε «,<?.=. Φεooψ [D2.2]

T2.13 [φψ].'.U(φ)M(ψ): [φ]: Φε «>(? .=. Φε ooi//:D: [μ]: μ(φ) .=. μ(ψ)

PF: [<pιf/].\Hyp(3):z>:

4) [α]:oo<«> εooψ .=.°°<a> ε ooψ : [3]

5) [a]:φ{a}M(φ)=.ψ{a}M(ψ): [T2.6, 4]

6) [a]:φ{a}.=.ψ{a}: [5, 1 , 2 ]

[μ]: μ(φ)=.μ(ψ) [Tl.ll, 6]

T2.14 [^]::N(^).N(ι//).3.'.[φ]:ΦεOo^.=.Φεooι//:=: [μ]: μ(φ).=. μ(ψ)

[T2.12, T2.13]

The hypothesis of this thesis guarantees that for numerical predicates,

there is derivable a thesis analogous to the principle of ontological


extensionality for the primitive epsilon: compare the consequent of thisthesis with T1.5. Next, identity for numerical names is defined and a thesisanalogous to the principle of extensionality for identical names is derived.

D2.3 [ΦΨ]:Φεoo*.*εooΦ.=.Φ = oo*T2.15 [ΦΦJΛΦ = «,*.=>: [μ]:μ(φ).=. μ(¥)

PF: [Φ*]ΛHyp(l).D:2) Φεoo^.ΦεooΦ. [D2.3, 1]3) N(Φ).N(Φ): [T2.2, 2]

4) [X]:XεooΦ.=.XεooΦ: [T2.4, 2][μ]:μ(Φ) =. μ(Φ) [T2.14, 3, 4]

T2.16 [ΦΦ]. ' . [μ]: μ(Φ).= . μ(Φ):Φε 0oΦ.Φε 0 OΦ:Z).Φ = 0OΦ

PF: [Φ*]ΛHyp(3):D:

4) N(Φ) [T2.2, 2]5) N(Φ): [Γ2.£, 3]6) [X]:XεooΦ.=.XεooΦ: [Γ2.24, 1, 4, 5]7) Φ ε w Φ . [6, 2]8) ^εooΦ. [6, 3]

Φ=ooΦ [D2.3, 7, 8]

Γ2.27 [ Φ ^ ] . \ Φ = oo^. Ξ :ΦεooΦ.^ε o o Φ: [μ]: μ(Φ) .=. μ(Φ)[Γ^.25, Γ2.25, D^.5]

It should be noted that in dealing with identical numerical names, thecondition imposed on the principle of numerical extensionality is alwaysfulfilled. Thus, this exact analogue of the principle of extensionality foridentical names is derivable, compare T1.7.

This section is concluded by establishing that under certain conditionsa thesis analogous to an ontological definition is always forthcoming for anydefinable numerical name.

T2.18 [Φ^].*.ΦεooΦ.N(^):[«]:Φ{α}.3. <ρ{α}:D.Φεoo<ρ [T2.2]T2.19 [Φ<p].".Φεoo<ρ.=>: ΦεooΦ M{φ)\ [β]:Φ{α}.D. φ{a}

PF: [Φ(^]::Hyp(l) .=).'.

2) ΦεooΦ.Q(Φ).N(<p).\ [T2.2, 1]

[ 3 * ] . ' .3) Φ{b}.φ{b}: [T2.2,l]4) [a]:Φ{a}.D.booa: [D1.21, 2, 3]5) [a]:bcoa.o.φ{a}.\ [D1.20, 2, 3]6) [a]:Φ{a}.Ώ.φ{a}: [4, 5]

Φε^Φ.N(<p): [a]:Φ{a}.Ώ.φ{a} [2, 6]

T2.20 [Φ^].'.Φεoo^.=:ΦεooΦ.N(<^):[α]:Φ{α}.D.^{α} [T2.18, T2.19]T2.21 [μΦφψ].'. [φa]: μ < φ>(°°<a>) .= . ψ< φ>{a}\ Φzoo'ψ<φ>\^,

μ<φ>{Φ)PF: [μΦ^]::Hyp(2):D.\

3) [ab]:Φ{a} .Φ{b}.Ό.aoob: [T2.2, 2, D1.21]


4) [ab]: Φ{a}. aco b .Ό.Φ{b}: [T2.2, 2, D1.20]

[3α] .\5) Φ{a}.ψ<φ>{a}: [T2.2, 2]6) [b]:Φ{b}.Ώ.aoob: [3, 5]7) [b]:acob.z>.Φ{b}: [4, 5]8) [b]:Φ{b}.=.°o<a>{b}: [D1.19, 7, 6]9) μ<φ>(Φ)=.μ<φ>(«><a».m. [Tl.U, 8]

μ<φ>(*) [1, 5, 9]

Γ2.22 [μψ].'. [φa]: μ< φ> (°o<α>) .=. ψ< φ>{a}:^: [Φφ]:Φε0Oψ<φ> .^>.μ<φ>{Φ) [T2.21]

T2.23 [μΦφψ].'. [φa]: μ<φ>(^<a>) =.ψ< φ>{a}:μ<(p>(Φ).ΦεooΦ .Φ{a}:^.ψ<φ>{a}

PF: [μΦφψ].'. Hyp(4):=>:5) Q(Φ).N(Φ): [T2.2, 3]6) [b]:Φ{b}.^.aoob: [D1.21, 5, 4]7) [b]: αoo & . D . φ {6}: [2)1.20, 5, 4]8) [b]:Φ{b}.= .«><a>{b}: [D1.19, 7, 6]

9) μ < ^ > ( Φ ) . Ξ . μ < ^ > ( o o < β » : [Γl.ll, 8]

ψ<^>{fl} [9, 2, 1]

Γ .̂̂ 4 [μΦ(^τ].'. [ψa]: μ <ι//>(°o<α>) .=. τ<ψ>{β}: *ε ooΦ. μ<(^>(Φ):D:

[α]: Φ {a} .D. T< (?> {«} [Γ2.25]Γ .̂̂ 5 [μΦ^τ].'. [ψa]: μ<ψ>(oo<a> ) =. τ< ψ>{a}: N(τ< φ>).

μ<(^>(Φ).ΦεooΦ:=).ΦεooT<^> [Γ2.24, Γ2.^θ]Γ2.25 [μr] : : [i//α]: μ<Ψ>(oo<a>) =. τ<ψ>{a}:Ώ.'. [Φφ] . ' .N(τ< φ» . D :

Φ ε o o τ < ^ > . = . Φ ε o o Φ . μ<(^>(Φ) [Γ2.22, Γ2.3, Γ2.^5]

This thesis guarantees that given (a) a protothetical definition of afunctor and (b) that the functor is numerical, then a thesis is derivableanalogous to an ontological definition of the functor—analogous in the sensethat the thesis is expressible as a proposition which has the form of aproper ontological definition, compare the consequent of T2.26 with theform of ontological definition given in section 1.

Actually, T2.26 considers as the functor to be defined only one which isa single link numerical name forming functor (whose link is a propositionforming functor for a single name argument). However, an inspection ofthe above proof shows that similar theses can be established for numericalnames with any finite number of links, for instance,

T2.27 [μτ]::[φψa\: μ.<φψ>(oo<a>) =.τ<φψ>{a}:Ώ.\

[Φφψ].'.N(τ<φψ>) . 3 : ΦεooT<<ρψ> .Ξ,Φε w Φ . μ < ^ > ( Φ )[similar to T2.26]

T2.28 [μr] : : [a]: μ(°o<β>) .= . τ{α}:D.\ [φ] . ' . N(τ) .=): Φ ε ^ r =. Φε^Φ . μ(Φ)[similar to T2.26]

It should be noted that theses T2.26, T2.27, etc., provide an effectivemeans for obtaining theses which have the form of ontological definitions.


In general, once an ontological definition of a numerical name formingfunctor is decided upon, it is obtainable from an effectively given proto-thetical definition. An application of this process is given in the nextsection.

With the availability of ontological definitions for numerical names, aswell as extensionality and an ontological axiom, it is clear that thenumerical epsilon is completely analogous to the primitive epsilon. So, aslong as one restricts attention to numerical functors, all theses whichinvolve the primitive epsilon are also derivable for the numericalepsilon. And given this basis, the means is available for providing anontological development of Peano's arithmetic in the system.

3. Numerals as names Peano's axioms expressed with the numericalepsilon are given as T3.1, T3.3, T3.5, T3.6 and T3.15. The proof of T3.15is of particular interest since it requires the use of an ontological defini-tion for numerical names. In this development T3.5 does not follow banallyin the system as does its analogue T1.34: the antecedent of T3.5 isrequired to establish its consequent.

T3Λ OεooFin [T2.2, T1.24, T1.26, T1.27, T1.20, T1.22]T3.2 [Φ].\ [α]:Fin{α}.D. [3A].AεA.~(Aεa):Φε0oΈt]n:o.S<Φ>ε0o'Fin

PF: [Φ]/.Hyp(2):D:3) Q(Φ).N(Φ) [T2.2, 2]

[3*]:4) Φ{α}.Fin{β}. [T2.2, 2]

[*A].5) AεA.~(Aεa). [1,6]6) <fl\JA)-Aoa. [T1.10, 5]7) AεaUA. [T1.8, 5]8) Φ{(aUA)-A}. [T1.6, 6, 5]9) S<Φ>{αUi}. [D1.25, 7, 8]

10) FinjtflM}: [T1.21, 5, 4]S<Φ>εooFin [T2.2, 9, 10, T1.30, 3, T1.31, T1.22]

T3.3 [ φ ] : Φ ε o o F i n . D . S < Φ > ε o o F i n [T3.2, Ax Infin]

T3Λ [φ].~(0 = ooS<φ>) [T2.2, D2.3, T1.28]

On the basis of,

D3.1 [ΦΦ]: Φε .oΦ . Φε ̂ . ~(Φε oo )̂ .=. Φ * ^

we have the following.

T3.5 [Φ]:ΦεooFin.D.0*ooS<Φ> [T2.3, T3.3, T3.1, T3.4, D3.1, D2.3]T3.6 [ΦΦ]: Φε ooFin . Ψε ooFin . S < Φ > = O O S < ^ > . D . Φ = OO^

PF: jWj:Hyp(3).=>.4) Q(Φ).N(Φ). [T2.2, 1]5) Q(Φ).N(Φ). [T2.2, 2]6) S < Φ > ε o o S < ^ > . [D2.3, 3]


7) [3α] .S<Φ>{z} .S<*>{α}. [T2.2, 6]

8) [3 δ].Φ {&}.*{&}. [T1.29, 5, 7]

9) Φ ε ^ Φ . [T2.2, 8, 4, 5]

10) * ε o o Φ . [ 7 ^ , 8, 4, 5]

Φ = oo* [iλ2.3, 9, 10]

In order to obtain the last of Peano's axioms, an ontological definition

of the intersection of numerical names is required. On the basis of the

following definitions,

D3.2 [Φ<ρι//]:Φεoo<ρ.Φεooi//.=. Π 0 O < ^ > ( Φ )D3.3 [aφψ]:oo<a>εooφ .°°<a>ZooY .=. Π oo< φψ>{a}

the hypotheses of T2.27 can be obtained.

Γ3.7 [«(pi//]: Π 0 0< (^ι//>(oo<α>).Ξ. fλ oo< φψ> {a} [D3.2, D3.3]

T3.8 [abφψ]: noQ< φψ>{a}.a^b .D. Π ^ φψ>{b}

PF: M ^ ] : H y p ( 2 ) . D .

3) oo<a>εooφ.°°<a>εooψ. [D3.3, 1]

4) o o < δ > ε 0 0 ^ . o o < δ > ε 0 o i / / . [Γ^.7, 2, 3]

Πoo< <?•//>{&} [^>3.3, 4]

ΓJ.5 [ ^ ψ ] . N ( Π o o < ( ^ ψ » [D1.20, T3.8]

Now, on the basis of T2.27, the ontological definition of the intersection

of numerical names is derivable.

T3.10 [Φ^ι//]:ΦεooΠ00<(^ψ/>.Ξ.φεoo^.Φεooψ.ΦεooΦ [T2.27, T3.7, T3.9]

T3.ll [Φ^]:Φε0 0Π0 0<^ψ>.=.Φε0o^.Φεoo^ [T3.10, T2.3]

And given this thesis, Peano's last axiom follows.

T3.12 [Abφ].'.Oεooψ: [Ψ]: Ψε^φ . D . S<Ψ> ε ooψ-.AεA .φ{b}.~ (Aεδ):=).

φ{bUA}

PF: [Aδ^].".Hyp(5):D:

6) N(<^). [T2.2, 1]

7) °o<δ>εoo(p. [T2.6, 4, β]

8) S < o o < & » e o o ( ^ : [2, 7]

9) [a]:S<°o<b»{a}.Ώ.φ{a}: [T2.20, 8]

10) A ε δ u A . [ΓJ.δ, 3]

11) {bl)A)-Aob. [T1.10, 3, 5]

12) °o<b>{(buA)-A}. [T1.6, 11, ΓI.I5]

13) S<oo<δ»{δuA}. [i)i.25, 10, 12]

φ{bUA} [9, 13]

Γ3.i3 M . ' . O ε o o ^ : [φ]: ̂ ε oo(̂ . 3 . S< ^ > ε oo^ : D :

[Aδ]:AεA.<^{δ}.D.^{6uA} [T1.6, T1.9, T3.12]

T3.14 [Φ^]. ' .0εoo^: [Φ]: Φεoo<^.=>.S<Φ> εoo(^: Φε oo Fin : D . Φε ^φ

PF: [Φςp]::Hyp(3):D.'.

4) [Aδ]:AεA.^{6}.D. φ{buA}\ [T3.13, 1, 2]

5) N(<p). [Γ2.2, 1]


6) N(Φ).Q(Φ). [T2.2, 3]7) [lb].θ{b}.φ{b}. [T2.2, l ]

8) <?{Λ}.\ [T1.25, 7]

[3α] .\9) Φ{α}.Fin{α}: [T2.2, 3]

10) [Ab]:Aεb . <p{δ}.D. φ{bUA}: [T1.2, 4]

11) <?{«}•'• [i)2.25, 8, 9, 10]Φεoo^ [T2.2, 9, 11, 5, 6]

T3.15 [Φ<p].'.0ε «,</?: [Φ]: ̂ CooFin . #800 (? . D . S< Φ> εoo^rΦεooFin :D. Φε oo^PF: [Φ<p].\Hyp(3):D:

4) H(φ). [Γ^.2, 1]5) 0 ε o o Π o o < ^ F i n > : [T3.11, 1, Γ3.J]6) [ Φ ] : Φ ε 0 O n 0 O < ^ F i n > . D . S > < Φ > ε 0 o Π 0 o < ^ F i n > : [Γ5.il, 2, Γ5.3]7) Φ ε o o n o o < ^ F i n > . [T3.14, 5, 6, 3]

Φ ε ^ ^ [T3.U,Ί]

This thesis gives the principle of mathematical induction for finitenumerical names.

From an inspection of Peano's axioms as given in terms of thenumerical epsilon, the following equivalence could be expected.

T3.16 [Φ]: Φε^Φ .=.NC(Φ) [T2.2, D1.22, D1.14]

That is, cardinal numbers are just numerical individuals. Moreover, asimilar correspondence obtains for inductive cardinals (T3.28). As pre-liminary theses, there are the following.

T3.17 [α].oo<α>εoooo<α> [T2.2, D1.19, T1.16, D1.20, D1.21]

T3.18 [α]:Fin{α}.s .oo<α>ε 0 O Fin[T2.2, D1.19, T1.16, T1.17, T1.18, T1.22]

T3.19 [Φ]:Nn(Φ).D.ΦεooFin [T1.37, T1.36, T3.16, T1.22, T2.20, D1.16]

Now, the protothetical definition of a numerical satisfier,

D3.4 [μa]: [3Φ-]. Φε^Φ . Φ{a}. μ(Φ) .=. stsf oo< μ>{a}

is used in conjunction with T1.28 in order to obtain the ontologicaldefinition of that functor {T3.25).

T3.20 [μα]:stsf oo<μ>{«}.=3. μ ( ° ° < α »PF: [μfl]::Hyp(l).D. .

[3Φ]. .2) ΦεooΦ.φfc}. μ(Φ): [D3.4, 1]3) [b]:Φ{b}.= .°o<a>{b}: [T2.2, 2, D1.20, D1.21, D1.19]

4) [μ]: μ(Φ) .= . μ ( o o < α » . " . [Tl.ll, 3]μ(oo<α>) [2,4]

T3.21 [μα]: μ ( o o < « » . D . s t s f 0 0 < μ > { « } [D3.4, T3.17, T1.16]

T3.22 [μβ]: μ(oo<α>).s.stsfoo<μ>{«} [T3.20, T3.21]T3.23 [μab ]: stsf o o<μ>{α}.«c 5oδ.D. stsf oo< μ > {b}


PF: [μαδ]:Hyp(2).D.

[3Φ]

3) ΦεooΦ.Φ{α}.μ(Φ). [D3.4, 1]

4) Φ{b}. [T2.2, 3, D1.20, 2]

stsfoo<μ>{δ} [D3.4, 3, 4]

T3.24 [μ].N(stsfoo<μ» [D1.20, T3.23]

T3.25 [Φμ]:ΦεooStsf oo<μ>.= .ΦεooΦ. μ(Φ) [T2.28, T3-24, T3.22]

Given this ontological definition of a numerical satisfier of a predicate

(compare D1.8), the desired equivalence is derivable.

T3.26 [Φμ]:Φε o o Fin. μ(0): | > ] : μ(Φ) . 3 . μ ( S < * > ) : 3 . μ(Φ)

PF: [Φμ].\Hyp(3):=>:

4) OεooStsfoo < μ > : [T3.25, 2, Γ3.2, Γ2.3]

5) [Φ]: *ε ooFin . * ε ^stsf ^ < μ > .D. S< * > ε ooStsf oo< μ > :

[T3.25, T3.21, T2.3, 3]

μ(Φ) [T3.15, 1, 4, 5, T3.25]

T3.27 [Φ]: ΦεooFin . D . Nn(Φ) [T3.26, D1.26]

T3.28 [Φ]: ΦεooFin.Ξ.NΠ(Φ) [T3.27, T3.19]

That is, natural numbers are just finite numerical individuals. The

equivalence of the axiom of infinity and one of Peano's axioms is now

shown.

T3.29 [a].'. [Φ]:ΦεooFin.D.S<Φ>εooFin:Fin{α}: [A]:AεA .D.Aεα D.

[lA\.AzA.~(Aza)

PF: [fl]/.Hyp(3):D:

4) α o < β > ε o o F i n . [T3.18, 2]

5) S < o o < e » C o o F i n : [1, 4]

[36]:6) S<oo<a»{b}.Fin{b}. [T2.2, 5]

[35].

7) Bεb .oo<a>{b-B}. [D 1.25, 6]

8) b-Baa. [D1.9, D1.12, T1.2, 3]

9) aob-B. [T1.23, 8, 2, D1.19, 7]

10) £ ε £ . [ΓJ.^, 7]

11) £εα. [10, 3]

12) - (5εJ5) : [ΓI.5, 9, D2.12, 11]

[3A].AεA .-(Aεα) [12, 10]

Γ3.3ί? [Φ]: ΦεooFin.D.S<Φ>ε o oFin:=): [a]: Fin{a}.D. [3A].AεA .~(Aεa)

[T3.29]

T3.31 [φ]:ΦεooFin.D.S<Φ>εooFin:=.AxInfin [T3.30, T3.3, Axlnfin]

Naturally, theses analogous to the ontological definitions of arithmeti-

cal operations could be given and the properties of the operations are all

derivable on the above basis: the requisite definitions for the operations

having already been given in the introductory section. For example, in


respect to addition, one has uniqueness, closure, and the recursiveproperties of the operation,

T3.32 [Φ(pψ].\ΦεooFin.D:Φ = oo ̂ + ψ.=. Sm(Φ^ψ)T3.33 [ΦΦ]: Φε ooFin . * ε ooFin . 3 . [3 X]. Xε ^Fin . X = „ Φ + ΦT3.34 [ Φ ] : Φ ε o o F i n . = ) . Φ = OoΦ + 0T3.35 [ΦΦ]:ΦεooFin.*εooFin.D.Φ + S<Φ>=ooS<Φ + * >

and similarly for any other arithmetical operation which can be definedrecursively.

4. Concluding remarks In section 1, as in section 3, an analysis of thecardinality of (finite) names is given. These sections differ in style: theformer treats numerals as predicates, the latter as names. Consider thefollowing.

T1.32 Nn(0)T3.1 OεooFin

Under the intended interpretation both of these propositions state that zerois a natural number. Naturally the numeral in these propositions belongsto but one semantical category, for the numeral is a unique constant (of thecategory of proposition forming functors for one name argument). But thefirst proposition corresponds in a simple way to,

*120.12. h.OeNC induct

of [6], while the second proposition does not.For the authors of [6], classes are but "fictitious objects" and their

use of an epsilon and class abstractor is but a notational device forspeaking extensionally about predicates. But the logic of section 1 iscompletely extensional, and this is achieved without the introduction of"fictitious objects" via notational devices. Section 1 gives, then, a develop-ment of cardinality closely akin to [6], but without the fictions of that work.

On the other hand, the numerals of section 3 can clearly be consideredto be names. Section 2, in providing an internal ontological model for thenumerical epsilon, justifies this claim. Recall that in section 2 analoguesof the ontological axiom, extensional theses and definitional theses of theprimitive epsilon were derived for the numerical epsilon. This guaranteesthat an analogue of each thesis derivable for the primitive epsilon isderivable for the numerical epsilon. In this exposition, unlike [6], there isno question of numerals being "really" predicative and only "conven-tionally" nominative—since there is an internal ontological model for thenumerical epsilon, its arguments are by analogy nominal.

Thus, if one wishes to construe numerals as predicates there is theexposition of section 1 without the disadvantages of [3], while if one wishesto construe numerals as names, there is the exposition of section 3. In thelatter case there is, as has been seen, formal justification for consideringnumerals as names, while in the former case, one can at best considerthem fictitious names—they are indeed only predicates.


Though an analysis of the cardinality of names has been given, what islacking in this exposition is an analysis of cardinality in general. There is,for instance, a definition of a constant number (zero) for names, butnowhere has the definition of the constant number zero for propositionforming functors of one name argument (or two, or three name arguments,etc.) been given. Only the cardinality for the primitive semantical categoryof names has been investigated, while the cardinality of the non-primitivecategories remains to be discussed.

Consider the following definition.

[Φφ].'. [3α]. Φ{α}. φ{a}\ [ab]: Φ{a}. Φ{b} .D.aob:=. Φε φ

Here a higher (ontological) epsilon is defined. With this definition slightadjustments in the exposition of section 2 would establish the existence ofan internal ontological model for this epsilon. Higher epsilons of this typeare well known, and it can be shown, in fact that a higher epsilon exists foreach and every semantical category definable in ontology.

Since there is a higher epsilon for each category of ontology, theexposition for the cardinality of names which has been given serves as anexposition for the cardinality of any particular category of ontology. Thecardinality of any given category is to be developed in exactly the mannerin which the cardinality of names has been given, employing the higherepsilon peculiar to the category in question in place of the primitiveepsilon.

It is important to note that the axiom of infinity is reproducible athigher levels. It is already clear that the ontological directives and axiomare available for any category, but the axiom of infinity must also be avail-able. However, it can be shown that the axiom of infinity for propositionforming functors of one name argument is implied by the axiom of infinityfor names, and in general, that the axiom of infinity for any non-primitivesemantical category ultimately definable in terms of the primitive categoryof names is implied by the axiom of infinity for names.

At this point a precise exposition of arithmetic for each category ofontology has been attained. No doubt this is a far cry from the standpointof ordinary arithmetic which ignores differences of type in numbers, but onthe basis of these different types of arithmetic one can safely reach thestandpoint of ordinary arithmetic.

The only remaining requirement in the quest for arithmetic is theability to handle arithmetical operations whose arguments are non-homogeneous. The possibility already exists of producing, for instance, thesum of homogeneous cardinals. But consider the problem of adding thecardinality of a name to the cardinality of some other semantical category.Since the operation of addition has homogeneous arguments, it is necessaryto secure elements of the same category in order to perform the operation.But this can always be done in a satisfactory manner by producing anelement of the higher category equinumerous with the element of the lowercategory.


Of course, this requires expanding the notion of equinumerosity so thatit countenances non-homogeneous arguments. But the higher epsilonsafford a precise basis for such an extension of the notion. It can be seenthat it is possible in the presence of the higher epsilons, to performarithmetical operations, now, for arguments from any semantical categorywhatsoever. One simply finds a representative from the higher categoryfor the element of the lower category and performs the operation in thearithmetic of the higher level.

And it is now clear in what sense the standpoint of ordinary arithmeticwhich ignores the difference in type among numbers can be attained inontology. A sufficiently high semantical category is selected and all workis carried on in the arithmetic of that level. It is not that numbers areambiguous as to type, as the authors of [6] would have it, but that any givenarithmetical problem can be handled in a single sufficiently high semanticalcategory.

At this point, the work of deriving arithmetic is finished. Indeed, theabove discussion closely parallels section *126 of [6] which discusses"typically indefinite inductive cardinals " . The advantage to this expositionlies in the fact that the standpoint of ordinary arithmetic has been achievedwithout any of the ambiguities latent in the development of [6]. By relyingon the theory of higher epsilons, the effect of employing typically ambigu-ous symbolism as in [6] can be obtained in ontology without howeverexhibiting theorems which are ambiguous.

Finally, having supplied an ontological model for Peano's arithmetic inontology (extended by an axiom of infinity), the incompleteness of thissystem will follow, if its directives are recursive. In this respect, thenumerical epsilon proves very useful. The directives for ontology weregiven by Lesniewski [3] in a list of terminological explanations which aredeveloped by employing his mereological concepts. Now, the numericalepsilon provides an efficient means of modeling Lesniewski's originalterminological explanations in Peano's arithmetic as given in section 3,thus showing the applicability of Gόdel's incompleteness result to ontology.The details of this work are left for another paper.

REFERENCES

[1] Canty, John Thomas, Leέniewski's ontology and GόdeVs incompleteness theorem,Ph.D. Thesis, June, 1967, University of Notre Dame, Notre Dame, Indiana.

[2] Lejewski, Czesiaw, 'On LesΊiiewski's ontology," Ratio, v. 1 (1957), pp. 150-176.

[3] Lesniewski, Stanisiaw, "Grundzϋge eines neuen Systems der Grundlagen derMathematik,'' Fundamenta Mathematicae, v. 14 (1929), pp. 1-81. "Uber dieGrundlagen der Ontologie," Comptes rendus des seances de la Sociέtέ des sci-ences etdes lettres de Varsovie, Classe III, v. 23 (1930), pp. 111-132.

[4] Siupecki, Jerzy, "St. LesΊiiewski's protothetics," Studia Logica, v. 1 (1953), pp.44-112 and "St. Les*niewski's calculus of names," v. 3 (1955), pp. 7-71.


[5] Sobociήski, Boleslaw, "On the single axioms of protothetic," Notre Dame Journalof Formal Logic, v. 1 (1960), pp. 52-73 and v. 2 (1961), pp. 111-126, 129-148 and"On successive simplifications of the axiomatizations of Les*niewski's ontology,"(in translation) Polish Logic, Storrs McCall (Editor), Oxford University Press,1967, pp. 188-200.

[6] Whitehead, Alfred N., and Bertrand Russell, Principia mathematica, Vol. I-III,(second edition), The University Press, Cambridge, 1963.

University of Notre DameNotre Dame, Indiana

john thomas canty - projecteuclid.org

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